Dynamical Equations Of Periodic Systems Under Constant External Stress

1. Dynamical Equations Of Periodic SystemsUnder Constant External Stress Gang Liugang.liu@queensu.caHPCVL, Queen's University, Kingston, ON 2014 CAP Congress, LaurentianUniversity, SudburyJune 2014

2. PURPOSE To derive the concrete mathematical form of dynamical equations for the periodvectors of a periodic system under constant external stress, from Newton's Second Law. http://arxiv.org/pdf/cond-mat/0209372.pdf

3. Classical Molecular Dynamics (MD) widely used in many fields. Periodic boundary conditions are often employed, then the system becomes a dynamical crystal filled with repeating cells.

4. In this work, the whole system is modeled as a limitedmacroscopicbulk, composed of unlimited number of repeated microscopiccells in three dimensions, with surface effect ignored.

5. As usual, the cell in the center is called MD cell. Particles in it called MD particles with position vectors , .

6. For each cell, the three edge vectors (forming a right-handed triad) are the period vectors of the system.

7. Degrees of freedom of the system. Then the MD particle position vectors and the period vectors are the full degrees of freedom of the system.

8. What about the periods? The dynamics of the particles is just Newton's Second Law on them.

9. Another respect: external forces crystal

10. Another respect: external forces External forces will definitely cause its internal structure to change. external forces crystal

11. Another respect: external forces External forces will definitely cause its internal structure to change. external forces crystal

12. In 1980, Parrinello and Rahman proposed their theory of the period dynamics (PRMD), where they introduced a Lagrangian and brought it into the Lagrangian Dynamical Equation to produce dynamics for both the MD particles and the periods. and

13. Equations of PRMD (constant external pressure) with

14. PRMD combined with the well-known Car-Parrinello MD later, has been used extensively in many kinds of simulations.

15. In 1983, Nose and Klein pointed out in the paper This implies that the generated dynamical equations for the MD particles in PRMD are not that of Newton’s Second Law.

16. Another drawback in PRMD The generated dynamical equation for the periods under constant external stress in their paper . It is not in a form where the periods are driven by the imbalance between the internal and external stresses. Then when the systemreaches an equilibrium state, the internal and external stresses may not balance each other.

17. In any case, PRMD can find the true equilibrium states under constant external pressure and zero temperature, when all velocities and accelerations are zero.

18. Not using Lagrangian Dynamics, we will Keep Newton’s Second Law for the MD particle as its original Apply Newton’s Second Law on halves of the system and statistics over system translation and particle moving directions to derive dynamical equations of the periods.

19. As a result, our dynamical equations are in the form where the periods are driven by the imbalance between internal and external stresses. The internal stress has both a full interaction term and a kinetic-energy term.

20. Some notations By using the periods , any cell can be represented with where are any integers. For the MD cell . Cell volume Cell surface vectors:

21. More notations Only pair potential is considered Force acting on particle in cell by particle in cell is denoted with External stress with external pressure as a special case where is a unit matrix.

22. Net force on the MD cell Since for any action from the green cell on the red MD cell, the reaction is equal to the action from the blue on the red, we have

23. Now let us first cut the system into two parts with plane , so that for a given period , the right part contains cells of , the rest in the left part.

24. A better illustration The red is the MD cell.

25. The net external force on is the cross section vector in plane

26. Newton’s Second Law on is the net force on by

27. Newton’s Second Law on

28. Newton’s Second Law on is evenly distributed cell by cell in

29. Newton’s Second Law on “half cell bar”

30. is the net force of blues on red and greens The red is the MD cell.

31. The net force of blues on red is the negative of that of red on blues. The red is the MD cell.

32. The net force of blues on green is the negative of that of red on blues. The red is the MD cell.

33. The net force of blues on green is the negative of that of red on blues. The red is the MD cell.

34. The net force of blues on green is the negative of that of red on blues. The red is the MD cell.

35. Newton’s Second Law on

36. is the net force of blues on red and greens The red is the MD cell.

37. The net force of blues on red is equal to that of red on greens. The red is the MD cell.

38. The net force of blues on green is equal to that of red on greens. The red is the MD cell.

39. The net force of blues on green is equal to that of red on greens. The red is the MD cell.

40. The net force of blues on green is equal to that of red on greens. The red is the MD cell.

41. Newton’s Second Law on

42. Recalling potential energy of the MD cell

43. Introducing main interaction tensor/dyad

44. Remembering in

45. The left side of where and and is used.

46. The left side of where Since in , any is non-negative, and for any , there exists to cancel it, then is zero.

47. First form of the period dynamics directly from simplifying Newton’s Second Law

48. Now let us consider two states Only difference is translation between them.

49. Since they are indistinguishable We should take an unweighted average of the dynamical equations over all such states.

50. In all such states, only different